The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.

— Carl Gustav Hempel

First sentence of 'Geometry and Empirical Science', collected in Carl Hempel and James H. Fetzer (ed.), The Philosophy of Carl G. Hempel: Studies in Science, Explanation, and Rationality (2001), Chap. 2, 18. Also Carl Hempel, 'Geometry and Empirical Science', collected in J.R. Newman (ed.), The World of Mathematics (1956), Vol. 3, 1635.

The propositions of mathematics have, therefore, the same unquestionable certainty which is typical of such propositions as All bachelors are unmarried, but they also share the complete lack of empirical content which is associated with that certainty: The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter.

— Carl Gustav Hempel

From 'On the Nature of Mathematical Truth', collected in Carl Hempel and James H. Fetzer (ed.), The Philosophy of Carl G. Hempel: Studies in Science, Explanation, and Rationality (2001), Chap. 1, 13. Also Carl Hempel, 'On the Nature of Mathematical Truth', collected in J.R. Newman (ed.), The World of Mathematics (1956), Vol. 3, 1631.

To characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental.

— Carl Gustav Hempel

From 'Geometry and Empirical Science', collected in Carl Hempel and James H. Fetzer (ed.), The Philosophy of Carl G. Hempel: Studies in Science, Explanation, and Rationality (2001), Chap. 2, 24. Also Carl Hempel, 'Geometry and Empirical Science', collected in J.R. Newman (ed.), The World of Mathematics (1956), Vol. 3, 1641.

In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion.
(1987) -- Carl Sagan